Ample generics in automorphism groups of Boolean powers of simple Mal'cev algebras

Abstract

Let $\mathbf{A}$ be a finite simple Mal'cev algebra, such as for example a finite simple group, module, ring, associative or Lie algebra, loop or quasigroup. We show that the automorphism group of a filtered Boolean power of continuous functions from the Cantor space $2^\omega$ to $\mathbf{A}$ has ample generics. The proof splits into the abelian and non-abelian cases. In the abelian case, we use a representation by modules and the theory of $n$-systems developed by Kechris and Rosendal. In the non-abelian case, the proof relies on the decomposition of the automorphism group as a semidirect product of a certain closure of a filtered Boolean power of continuous functions from $2^\omega$ to the automorphism group of $\mathbf{A}$ and the stabiliser of finitely many points in the homeomorphism group $\mathrm{Homeo}\, 2^\omega$. As an intermediate step, we show that pointwise stabilisers in $\mathrm{Homeo}\, 2^\omega$ have ample generics, which extends Kwiatkowska's result that $\mathrm{Homeo}\, 2^\omega$ has ample generics.